We use the asymptotic notation \(O(\cdot )\)  in the usual way
[1]}. In particular,
let \(X \subseteq \mathbb {R}^n\) ,
\(f \colon F \subseteq \mathbb {R} \times X \rightarrow \mathbb {R}_{+}\) ,
\(g \colon G \subseteq \mathbb {R} \rightarrow \mathbb {R}_{+}\) ,
\(H \colon F \times \mathbb {R}_{+} \rightarrow \mathbb {R}_{+}\)  and
\(a \in \mathbb {R}\)  be such that
\(a = \lim _{i \rightarrow \infty } s_i\)  for some sequence
\((s_i,x_i)_{i \in \mathbb {N}}\)  in \(F\) , and suppose that
\(s \in G\)  whenever \((s,x) \in F\) .
Then \(f(s,x) \le H(s,x,O(g(s)))\)  as \(s \rightarrow a\) , uniformly in \(x\) , if
there exist \(k \colon G \rightarrow \mathbb {R}_{+}\)  and a
neighborhood \(U \subseteq \mathbb {R}\)  of \(a\)  such that
\(k(s) = O(g(s))\)  as \(s \rightarrow a\)  and
\(f(s,x) \le H(s,x,k(s))\)  whenever \((s,x) \in F \cap ( U \times X )\) ,
and similarly for \(a \in \lbrace  \infty , -\infty \rbrace \) .
